```{-
Copyright (C) 2011 Dr. Alistair Ward

This program is free software: you can redistribute it and/or modify
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program.  If not, see <http://www.gnu.org/licenses/>.
-}
{- |
[@AUTHOR@]	Dr. Alistair Ward

[@DESCRIPTION@]	Defines the classes of /Pi/-algorithm which have been implemented.
-}

module Factory.Math.Pi(
-- * Type-classes
Algorithmic(..),
-- * Types
-- ** Data-types
Category(..)
) where

import qualified	Data.Ratio
import qualified	Factory.Math.Precision	as Math.Precision
import qualified	ToolShed.Defaultable

{- |
* Defines the methods expected of a /Pi/-algorithm.

* Most of the implementations naturally return a 'Rational', but the spigot-algorithms naturally produce a @[Int]@;
though representing /Pi/ as a big integer with the decimal point removed is clearly incorrect.

* Since representing /Pi/ as either a 'Rational' or promoted to an 'Integer', is inconvenient, an alternative decimal 'String'-representation is provided.
-}
class Algorithmic algorithm where
openR	:: algorithm -> Math.Precision.DecimalDigits -> Data.Ratio.Rational	-- ^ Returns the value of /Pi/ as a 'Rational'.

openI	:: algorithm -> Math.Precision.DecimalDigits -> Integer			-- ^ Returns the value of /Pi/, promoted by the required precision to form an integer.
openI _ 1	= 3
openI algorithm decimalDigits
| decimalDigits <= 0	= error \$ "Factory.Math.Pi.openI:\tinsufficient decimalDigits=" ++ show decimalDigits
| otherwise		= round . Math.Precision.promote (openR algorithm decimalDigits) \$ pred decimalDigits

openS	:: algorithm -> Math.Precision.DecimalDigits -> String			-- ^ Returns the value of /Pi/ as a decimal 'String'.
openS _ 1	= "3"
openS algorithm decimalDigits
| decimalDigits <= 0	= ""
| decimalDigits <= 16	= take (succ decimalDigits) \$ show (pi :: Double)
| otherwise		= "3." ++ tail (show \$ openI algorithm decimalDigits)	--Insert a decimal point.

-- | Categorises the various algorithms.
data Category agm bbp borwein ramanujan spigot
= AGM agm		-- ^ Algorithms based on the /Arithmetic-geometric Mean/.
| BBP bbp		-- ^ <http://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula>.
| Borwein borwein	-- ^ <http://en.wikipedia.org/wiki/Borwein%27s_algorithm>.
| Ramanujan ramanujan	-- ^ <http://www.pi314.net/eng/ramanujan.php>.
| Spigot spigot		-- ^ Algorithms from which the digits of /Pi/ slowly drip, one by one.

instance (
ToolShed.Defaultable.Defaultable agm,
ToolShed.Defaultable.Defaultable bbp,
ToolShed.Defaultable.Defaultable borwein,
ToolShed.Defaultable.Defaultable ramanujan,
ToolShed.Defaultable.Defaultable spigot
)  => ToolShed.Defaultable.Defaultable (Category agm bbp borwein ramanujan spigot)	where
defaultValue	= BBP ToolShed.Defaultable.defaultValue

instance (
Algorithmic agm,
Algorithmic bbp,
Algorithmic borwein,
Algorithmic ramanujan,
Algorithmic spigot
) => Algorithmic (Category agm bbp borwein ramanujan spigot)	where
openR algorithm decimalDigits
| decimalDigits <= 0	= error \$ "Factory.Math.Pi.openR:\tinsufficient decimalDigits=" ++ show decimalDigits
| decimalDigits <= 16	= Math.Precision.simplify (pred decimalDigits) (pi :: Double)
| otherwise		= (
case algorithm of
AGM agm			-> openR agm
BBP bbp			-> openR bbp
Borwein borwein		-> openR borwein
Ramanujan ramanujan	-> openR ramanujan
Spigot spigot		-> openR spigot
) decimalDigits

openI _ 1				= 3
openI (Spigot spigot) decimalDigits	= openI spigot decimalDigits
openI algorithm decimalDigits
| decimalDigits <= 0	= error \$ "Factory.Math.Pi.openI:\tinsufficient decimalDigits=" ++ show decimalDigits
| otherwise		= round . Math.Precision.promote (openR algorithm decimalDigits) \$ pred decimalDigits

```